Thermal spin-orbit torque in spintronics

Based on the spinor Boltzmann equation (SBE) formalism, we present a theory of temperature-dependent thermal spin-orbit torque for a system in the presence of Rashba spin-orbit interaction. Under the local equilibrium assumption, we can expand the distribution function of spinor Boltzmann equation around local equilibrium distribution; then, the spin diffusion equation can be derived from SBE, where the spin transfer torque, spin orbit torque as well as thermal spin-orbit torque can be naturally obtained. It is shown that this thermal spin-orbit torque originates from the temperature gradient of local equilibrium distribution function, which is explicit and straightforward than previous works. Finally, we illustrate them by an example of spin-polarized transport through a ferromagnet with Rashba spin-orbit coupling, in which those torques driven whatever by temperature gradient or bias are manifested quantitatively.